Trig Identities from Properties of Complex CalculationsFormulas for cos(2A) and sin (2A)From cis(2A) = cis(A)*cis(A) we found
Therefore comparison of rectangular coordinates/components (or real and imaginary parts) yields the double angle formulas
Formulas for cos(nA) and sin (nA), the case n =3.Observe
Therefore
Equality of corresponding real and imaginary parts gives
Exponential or cis Expressions for Trig FunctionsFrom the two equations exp(iA) = [cos(A),sin(A)] = cos(A) + i sin(A) exp(-iA) = [cos(-A),sin(-A)] = cos(A) - i sin(A) we see exp(iA) + exp(-iA) = 2 cos(A) and exp(iA) - exp(-iA) = 2i sin(A) Therefore (1/2)[exp(iA) + exp(-iA)] = cos(A) and (1/2i) [exp(iA) - exp(-iA)] = sin(A) Therefore tan(A) = sin(A)/cos(A) = (1/i) { [exp(iA) - exp(-iA)]/[exp(iA) + exp(-iA)] } or tan(A) = sin(A)/cos(A) = -i{[exp(iA) - exp(-iA)]/[exp(iA) + exp(-iA)]} So all trig functions may be expressed in terms of exp(iA) and exp(-iA). The substitution of exp(iA) and exp(-iA) expressions for trig functions turns the proof or derivation of trig identities into simple algebraic exercises involving complex numbers and the add the angles rule for multiplication of exp(iA) with exp(iB). While highschool students may be taken through the exercises of proving trig identities before meeting complex numbers, the above quick explanation of complex numbers and its links to trig imply a quick route through highschool mathematics courses on algebra and trig. In retrospect, the presentation of trig in highschool has been harder than need-be. More motivation and more applications of trig, tied to right triangles and complex numbers, come from engineering and physics. The description of electric currents and devices depends on sine and cosines, or more simply, if you know complex numbers and exp(iA) = exp(iA). The latter functions also appear in the theoretical treatment of statistics and of higher mathematics (analysis). . Exponentials of Complex NumbersIf you have met the exponential ea = exp (a) of a real number a then you may be please to know that the exponential of a complex number a + ib (where a and b are real) is given by
Take this as a computational definition. Higher mathematics in science, engineering, statistics and economics is often best seen with a knowledge of the exponentials ea or eib or ea+ib . One may write ex instead of exp(x), or vice versa.
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The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers. Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision . D1 to D6 after provide a review of vectors. More on Complex Numbers:
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